Fix \(j\in \lbrace 1,\dots ,k\rbrace \)  and let \(S_j=\lbrace (x_i, y_i)\rbrace _{i=1}^{m_{j}}\)  be the training dataset from the \(j\) -th cluster (having \(m_j\)  points) with
\(x \in _{j}\)  and \(y \in _{j}\)  s.t. \(_{j} = \lbrace x: x \in C_j\rbrace \)  and \(_{j} = \lbrace y: y \in C_j\rbrace \) .
Then, the following theorem — a direct application of a previous result [1]}, [2]} to our problem — suggests a potential benefit of clustering in terms of the expected error \(_{x,y}[\ell (f(x),y)]\)  for unseen data:
